Standard functions on rational numbers
\begin{code}
-{-# OPTIONS -fno-implicit-prelude #-}
-
module Ratio
( Ratio
, Rational
-- Implementation checked wrt. Haskell 98 lib report, 1/99.
) where
+\end{code}
+
#ifndef __HUGS__
-import PrelNum
-import PrelNumExtra
-#endif
+
+\begin{code}
+import Prelude -- To generate the dependencies
+import PrelReal -- The basic defns for Ratio
+\end{code}
+
+%*********************************************************
+%* *
+\subsection{approxRational}
+%* *
+%*********************************************************
+
+@approxRational@, applied to two real fractional numbers x and epsilon,
+returns the simplest rational number within epsilon of x. A rational
+number n%d in reduced form is said to be simpler than another n'%d' if
+abs n <= abs n' && d <= d'. Any real interval contains a unique
+simplest rational; here, for simplicity, we assume a closed rational
+interval. If such an interval includes at least one whole number, then
+the simplest rational is the absolutely least whole number. Otherwise,
+the bounds are of the form q%1 + r%d and q%1 + r'%d', where abs r < d
+and abs r' < d', and the simplest rational is q%1 + the reciprocal of
+the simplest rational between d'%r' and d%r.
+
+\begin{code}
+approxRational :: (RealFrac a) => a -> a -> Rational
+approxRational rat eps = simplest (rat-eps) (rat+eps)
+ where simplest x y | y < x = simplest y x
+ | x == y = xr
+ | x > 0 = simplest' n d n' d'
+ | y < 0 = - simplest' (-n') d' (-n) d
+ | otherwise = 0 :% 1
+ where xr = toRational x
+ n = numerator xr
+ d = denominator xr
+ nd' = toRational y
+ n' = numerator nd'
+ d' = denominator nd'
+
+ simplest' n d n' d' -- assumes 0 < n%d < n'%d'
+ | r == 0 = q :% 1
+ | q /= q' = (q+1) :% 1
+ | otherwise = (q*n''+d'') :% n''
+ where (q,r) = quotRem n d
+ (q',r') = quotRem n' d'
+ nd'' = simplest' d' r' d r
+ n'' = numerator nd''
+ d'' = denominator nd''
\end{code}
+
+
+#endif
+